Formation and tuning of moir´e excitons in large-twist angle WS 2 /MoSe 2 heterobilayers

Moir´e lattices formed in twisted van der Waals bilayers provide a unique, tunable platform to realize coupled electron or exciton lattices unavailable before. While twist angle between the bilayer has been shown to be a critical parameter in engineering the moir´e potential and enabling novel phenomena in electronic moir´e systems, studies of moir´e excitons so far have focused on closely angularly-aligned heterobilayers. The twist-angle degree of freedom has been largely considered detrimental to the observation of moir´e excitons. Here we report robust moir´e excitons in bilayers of even large twist angles formed due to Umklapp scattering by the moir´e reciprocal lattice vectors, and we furthermore demonstrate twist-angle tuning of the properties of the moir´e excitons as a result of varying moir´e reciprocal lattice periods. We develop an intuitive analytical model to explain our results, and, from the twist-angle dependence, obtain the eﬀective mass of the interlayer excitons and the electron inter-layer tunneling strength, which are diﬃcult to measure experimentally otherwise. These ﬁndings pave the way for understanding and engineering rich moir´e -lattice induced phenomena in angle-twisted semiconductor van der Waals semiconductor heterostructures.

even large twist angles formed due to Umklapp scattering by the moiré reciprocal lattice vectors, and we furthermore demonstrate twist-angle tuning of the properties of the moiré excitons as a result of varying moiré reciprocal lattice periods. We develop an intuitive analytical model to explain our results, and, from the twist-angle dependence, obtain the effective mass of the interlayer excitons and the electron inter-layer tunneling strength, which are difficult to measure experimentally otherwise. These findings pave the way for understanding and engineering rich moiré -lattice induced phenomena in angle-twisted semiconductor van der Waals semiconductor heterostructures.
Atomically thin heterostructures created by stacking van der Waals materials mark a new frontier in condensed matter physics [1][2][3]. When two monolayer crystals of the same lattice symmetries overlay on each other, a moiré superlattice may form due to a small mismatch in their lattice constants or angular alignment [4,5]. The latter -the twist angle between the two layersprovides a powerful tuning knob of the electronic properties of the heterostructure. Seminal results have been obtained in twisted bilayer graphene, where superconducting and correlated insulating states are created by fine control of the twist angle [6][7][8][9]. In semiconductors, such as transition metal dichalcogenides (TMDC) heterobilayers, the moiré lattice has a period on the length scale of an exciton, thereby providing a unique opportunity to create coupled exciton lattices hitherto unavailable in any other systems. A wide variety of phenomena, tunable with the twist angle, may become possible, ranging from single quantum-dot arrays and topological bands to strongly correlated states [10][11][12][13][14].
To search for the effects of moiré lattices on excitons, split exciton states have been reported in TMDC bilayers with very small twist angles, showing localization of exciton states likely in moiré super-cells [15][16][17][18]. However, increasing the twist angle has led to suppression of measurable features of moiré excitons. In WS 2 /MoSe 2 heterobilayers, it was suggested that the resonant interlayer hybridization amplifies the moiré superlattice effects on the electronic structure [19]; yet only a single resonance was resolved as the twist angle deviates significantly from 0 • or 60 • [18].
Existence of moiré superlattice for exciton in large-twist-angle bilayers and nontrivial effects of the twist-angle on excitons remain largely unexplored in experiments.
In this work, we show moiré excitons in heterobilayer of a wide range of twist angles and demonstrate tuning of their properties by the moiré lattice of varying periods. Utilizing the inter-and intra-layer hybrid excitons in WS 2 /MoSe 2 bilayers, we reveal the formation of moiré reciprocal lattices with Brillouin zones of different sizes at different twist angles. We furthermore show how the moiré reciprocal lattices drastically change the properties of the moiré excitons, such as their resonance energies, oscillator strengths, and inter-/intra-layer mixing. The twist-angle dependence of the moiré exciton states are well-explained by an analytical theory model based on band-folding in the moiré lattice, which also consistently explain the dependence on the spin-orbit splitting of the conduction band, valley selection rules, atomic stacking orders and the lattice symmetries.
Comparing the experimental results with the model, we obtain the effective mass of the interlayer excitons, the inter-layer electron tunneling strength. These results showcase moiré -lattice tuning as a new route to uncover and tune fundamental properties of heterobilayer systems, and open the door to studies of moiré exciton physics in twisted bilayers beyond graphene.

RESULTS
The devices used in this work are WS 2 /MoSe 2 heterobilayers with different twist angles θ, capped by few-layer hexagonal boron nitride (hBN). Details of sample fabrication and calibration of θ have been described elsewhere [20,21] and provided in the Method section. Fig. 1a shows the optical microscope image of a heterobilayer, where the sharp edges of two monolayers are aligned. The twist angle is θ = 59.8 • ± 0.3 • , determined optically using polarization-dependent second-harmonic-generation measurements [20,21] (See Supplemental Material Fig.S1 and Fig.S2 for details).

Identification and analysis of inter-and intra-layer hybrid excitons
We first characterize exciton hybridization in closely aligned hetero-bilayers, with a twist angle θ ∼ 0 • or 60 • . In such bilayers, the Brillouin zones of the two layers closely overlap in momentum space to form nearly direct bandgaps for both the inter-and intra-layer transitions (top panels of Fig. 1b). At the same time, the hole band offset is large but the conduction band offset is small between WS 2 and MoSe 2 (middle panels of Fig. 1b). Therefore inter-layer electron tunneling is expected between states of the same spin and valley, which leads to hybridization between the corresponding intra-and inter-layer exciton transitions that share the same hole state (bottom panels of Fig. 1b).
Making use of the large difference in oscillator strength between spatially direct and indirect excitons, we can identify the formation of hybrid states via the reflectance contrast (RC) spectra: , where R sample and R sub are reflection spectrum taken from sample and substrate respectively (See Supplemental Material Fig.S3). The interlayer exciton has an oscillator strength two to three orders of magnitude weaker than that of the intra-layer exciton, due to separation of the electron and hole wavefunction [22,23], so it is typically too weak to be measurable in absorption or RC spectroscopy where the noise level is typically 1% or higher (Supplemental Material Fig.S4). However, when interlayer excitons hybridize with intra-layer ones via electron or hole tunneling, the hybrid states acquire an oscillator strength through the intra-layer exciton component. Therefore, we can identify the hybrid excitons via their spectral weight in the absorption spectra of the heterobilayer.
As shown in Fig. 1c, the MoSe 2 monolayer region of the device (as marked on Fig. 1a) shows a strong intralayer MoSe 2 A exciton resonance near 1.65 eV, while the WS 2 monolayer has no exciton resonances nearby. In the bilayer, stacking of the WS 2 layer is expected to lead to a red shift of MoSe 2 A exciton resonance [18] while also introduce an interlayer exciton transition, between an electron in WS 2 and a hole in MoSe 2 . The interlayer exciton has a negligible oscillator strength and should not be observable in RC. However, two clearly-resolved resonances appear in our bilayer, both with significant spectral weight (top two spectra in Fig. 1c). The same two resonances are also measured in photoluminescence (See Supplemental Material Fig.S5). We therefore identify them as the inter-and intra-layer hybrid states, the lower (LHX) and upper hybrid excitons (UHX).
Both LHX and UHX inherit an oscillator strength from their intra-layer component [18], with the ratio f LHX /f U HX controlled by their intra-layer exciton fractions, which in turn is controlled by the energy detuning δ = E IX − E X between the uncoupled inter-layer (E IX ) and intra-layer (E X ) resonances. Therefore f LHX /f U HX greater or less than one corresponds to positive or negative detuning δ. There are multiple pairs of intra-and inter-layer excitons that can hybridize. We focus on the transition region of MoSe 2 A exciton first and label these states as M oA excitons, of which the hole is always in the highest MoSe 2 valence band. Other pairs will be analyzed later.
As clearly seen in Fig. 1c, in the R-stacking bilayers (θ = 2. To analyze the results quantitatively, we first obtain the energies, E LHX and E U HX , and oscillator strengths of the hybrid states by fitting the RC spectra using the transfer matrix method, where the hybrid excitons are modeled as Lorentz oscillators (See Supplemental Material Fig.S3) [24,25].
The fitted spectrum agrees well with the data, as shown in Fig. 1c. Describing the hybrid modes with the coupled oscillator model, we have (See Supplemental Materials Sec.I for details). Thereby using the fitted E LHX,U HX and f LHX,U HX , we can obtain δ and J. As summarized in Fig. 1d, we obtain J ∼ 20 meV for both R-and Hstacking and δ R − δ H = 25.9 ± 0.5 meV [26], consistent with the spin-orbit splitting of WS 2 [27], confirming the hybrid states are formed by spin-conserved inter-layer electron tunneling.

Twist-angle dependence of moiré -lattice induced hybrid excitons
To study tuning of the hybrid excitons by the moiré lattice, we perform the same measurements and analysis as discussed above on 30 samples with different twist angles, and obtain how the exciton energies, oscillator strengths and inter-layer tunneling vary with the changing moiré lattice.
As shown in Fig. 2a, the M oA hybrid exciton doublets are clearly resolved for θ 0 up to 6 • , which would correspond to a tuning of the moiré lattice constant by nearly three-fold [28]. The spectral weights of the doublets evolve continuously with the twist angle, reflecting continuous increase of f LHX /f U HX and δ with θ 0 (middle panel of Fig. 2c). At the same time, the inter-layer coupling J decreases continuously (bottom panel of Fig. 2c). These observations show clearly moiré lattice induce hybridization and tuning of intra-layer and inter-layer excitons, as we explain below.
We illustrate in Fig. 2b the M oA exciton bands at different twist angles, corresponding to the six samples shown in Fig. 2a. The intra-layer MoSe 2 A exciton transition (red band) remains direct, with the band minimum at zero center-of-mass momentum q X ∼ 0, irrespective of the twist angle.
It is close in energy with the inter-layer exciton formed by a hole from the same MoSe 2 valence band but an electron from a WS 2 conduction band. This inter-layer exciton band has the band minimum also at zero center of mass momentum: q IX ∼ 0, when θ ∼ 0 • (θ 1 in Fig. 2a- Fig. 2a-b), neglecting the small lattice constant mismatch.
As the two lattices rotate relative to each other by θ (θ 2 to θ 5 in Fig. 2a-

Theoretical analysis of moiré -lattice induced hybrid excitons
To analyze our results more quantitatively, we develop an analytical microscopic theory based on the above understanding (See Supplemental Material Sec.II for details). Comparing it with the measured twist-angle dependence of the hybrid states, we obtain the key band parameters of the bilayer, including the interlayer exciton effective mass and interlayer coupling strength.
We first compare the measured detuning δ with θ 0 and the interlayer exciton kinetic energy. As discussed above, δ is given by: where δ 0 is the detuning at θ = 0 • or 60 • for bilayers close to R-and H-stacking, respectively. q 1 is equal to 4π/(3a M ), and a M is the moiré period approximated by a 0 / θ 2 0 + 2 , for a 0 the monolayer lattice constant and the lattice constant mismatch |a 0 − a 0 |/a 0 between the two layers. Equation.
(1), we find the inter-layer exciton total mass M IX to be (6.9 ± 3.2)m 0 and (1.41 ± 0.28)m 0 for R-and H-stacking heterobilayers, respectively, for m 0 the electron free mass.
From our microscopic theory, we can also estimate the conduction-band interlayer tunneling parameter w from the coupling strength J through the relation where φ k and ψ k are respectively the relative-motion wave function for interlayer and intralayer Using our experimentally measured value of J at θ 0 ∼ 0, we estimate the interlayer tunneling w to be about 14 meV for both R-and H-stacking bilayers.
When the twist angle θ 0 is greater than 6 • , the hybrid exciton doublets become hard to be resolved, likely because there is a large blue detuning and the UHX has a vanishing oscillator strength (See Supplemental material Fig. S6 ).

Moiré excitons in commensurate moiré lattices at twist angles near 21.8 • and 38.2 •
Remarkably, pronounced and well-resolved doublets re-appear in hetero-bilayers with θ = 20.1 • ± 0.3 and 40.3 • ± 0.3, as shown in Fig. 3. In the bilayer with 20.1 • twist angle, the LHX has a smaller spectral weight than UHX has, corresponding to a negative detuning (δ = −5.6 meV), which is similar to H-stacking bilayers formed at θ ∼ 60 • . In contrast, in the bilayer with 40.3 • twist angle, the LHX has a larger spectral weight than UHX has, corresponding to a positive detuning (δ = 11.8 meV), which is similar to R-stacking bilayers formed at θ ∼ 0 • . In both devices, the coupling strength J ∼ 8 meV is weaker than but of the same order of magnitude as aligned bilayers with θ close to 0 • or 60 • .
The revival of hybrid excitons in these two bilayers can be understood as a direct result of interlayer tunneling induced by a moiré lattice that is nearly commensurate with the monolayer lattices. The two twist angles are close to the commensurate angles 21.

Moiré excitons formed with different intra-layer excitons
In the above discussion, we have focused on hybrid states formed with the MoSe 2 A excitons, which feature large spectral weight, relatively narrow linewidths, and well-resolved doublets at small detunings. Hybrid states can also form with higher-energy bands, including the MoSe 2 B, From our measurements of M oA and W A states in bilayers with θ < 1 • , we estimate ∆E R B of 10 to 16 meV (Fig. 4b). The value is significantly lower than predictions based on first principle calculations when interlayer tunneling is neglected [29,30]. Optical measurements.
For low temperature measurements, the sample is kept in a 4 K cryostat (Montana Instrument).
The excitation and collection are carried out with a home-built confocal microscope with an objective lens with numerical aperture (NA) of 0.42. For reflection contrast measurement, white light from a tungsten halogen lamp is focused on the sample with a beam size of 10 µm in diameter.
The spatial resolution is improved to be 2 µm by using pinhole combined with confocal lenses.
The signal is detected using a Princeton Instruments spectrometer with a cooled charge-coupled camera.
Data availability Data are available on request from the authors.
Competing interests The authors declare that they have no competing financial interests.